Pointing a bone towards an axis (in pose mode)
While you ask for a way without constraints, I think you will need to use one, at least temporarily, to get the bone positioned.


Align an empty to the head of the bone.
Move the empty on the global axis you want to align to.
Add a TrackTo constraint to the bone and set the empty as the target.


Now that the bone is pointing the direction you want, you can use &#9096; CtrlA->Apply Visual Transform to Pose and delete the constraint.

You can also use I->Visual Rotation to keyframe the values used after constraints have been applied to the bone.